George Altmann (University of Leeds) – Higher homotopy invariants of welded knots

Classical knot theory associates to a combinatorial knot diagram a knot embedding in the 3-sphere. The fundamental group (and peripheral system) of the complement give rise to powerful, combinatorially computable invariants. This talk explores an analogous construction for welded knots, a diagrammatic extension of classical knot theory corresponding to ribbon knotted tori in 4-space via the tube map.

We introduce a new topological invariant, the fundamental \pi-module (built using the first and second homotopy groups), and show how it can be computed combinatorially from a diagram. We then define a natural topological generalisation of the peripheral system, using the free loop space of the complement, and again show how this admits a simple combinatorial description.